Dot product
In mathematics, the dot product or scalar product[note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).
"Scalar product" redirects here. For the abstract scalar product, see Inner product space. For the product of a vector and a scalar, see Scalar multiplication.
Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle between two vectors is the quotient of their dot product by the product of their lengths).
The name "dot product" is derived from the dot operator " · " that is often used to designate this operation;[1] the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector (as with the vector product in three-dimensional space).
In physics, vector magnitude is a scalar in the physical sense (i.e., a physical quantity independent of the coordinate system), expressed as the product of a numerical value and a physical unit, not just a number. The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system. For example:[10][11]
Generalizations[edit]
Complex vectors[edit]
For vectors with complex entries, using the given definition of the dot product would lead to quite different properties. For instance, the dot product of a vector with itself could be zero without the vector being the zero vector (e.g. this would happen with the vector ). This in turn would have consequences for notions like length and angle. Properties such as the positive-definite norm can be salvaged at the cost of giving up the symmetric and bilinear properties of the dot product, through the alternative definition[12][2]
![{\displaystyle \mathbf {a} =[1\ i]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f68c1771ca29419b14a5e2334f03687f6e2670d6)
Computation[edit]
Algorithms[edit]
The straightforward algorithm for calculating a floating-point dot product of vectors can suffer from catastrophic cancellation. To avoid this, approaches such as the Kahan summation algorithm are used.
Libraries[edit]
A dot product function is included in: